Where do atheists get their morality from?

[Boccaccio]

I think most of the people here have argued that morality is subjective and has its origins with individual thinking and evolutionary predispositions. This isn't the result I expected with my OP. I believed most atheists felt that there are overriding moral principles that could be called objective.

As discussed, I think that equivocation over objective is unhelpful. Ordinarily, and you are doing it here, theists set up a false dichotomy where if you are not taking it from god (and theists usually avoid that part of the question, see stephenmills1000) then the world is incapable of being ordered in any fashion. One might assume you think god is necessary to push tennis balls back to earth or for birds to flock in flight or fish to shoal, yet all of these are explicable by relatively simple behaviour rules without the fish checking up with god on which way to turn.

I have trouble even writing this first question in a coherent way because the term "better" calls for a subjective judgement, but here it is. If morality is subjective how can one person's view or one group's views be better than another?

By better fulfilling species moral objectives. Have you read Sam Harris on that? I personally think he pays insufficient attention to moral objectives themselves but his discussion of measurable and predictable strategies for achievement of them is sound.

If our morality is in a large part defined by evolution, how can we trust ourselves to make a subjective moral judgement. Isn't moral judgement then just the reflexive neural response of an electromechanical ape-like meat machine?

See above. Other than your god-belief, what makes you think you are not a deterministic ape-like meat machine?

How can anyone claim the right to pass judgement on anyone else, since all morality would seem to have an equal footing?

You may argue the relative value of moral objectives but for given objectives we can in many cases measure the success of strategies toward them, so the "equal footing" comment is no more pertinent than saying that if a god does not tell us to drink fresh water why would we not drink sea water? Because it does not work so well for our longer term survival.

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Because I am a Christian I assume you know I believe in an external source for objective morality.

As I intimated earlier, I can readily provide you with an objective moral system not dependent on god. You will not like it, but it is objective as you define it, not relying on a human mind other than to accept that the moral rules. That is but one reason Craig's moral argument fails but that is not the topic here.

I do have a problem with evolutionary explanations for morality, ... because watching the complete indifference of animals to pain and suffering in each other seems to contradict the idea that humans being social animals, explains why humans value caring for each other.

Perhaps you would like to check your animal research before making claims like that. At least as far back as the mid-sixties, there was experimental rather than anecdotal evidence for one animal caring for another to its own detriment. This may interest you. There are studies on rats and I think rabbits which should also be available, if I recall correctly. Your presumption fails and you may need to reconsider the feasibility of moral objectives and rational behaviours toward their achievement.